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Journal of Physics C Solid State Physics

Critical exponents and many-body effects. IV

H P Griffiths, David W. Wood

Published Nov 21, 1974• 43 citations
Publisher

Abstract

For pt. III see abstr. A23213 of 1974. The critical exponent beta is examined for two three-dimensional Ising model systems which allow admixtures of three and four-body terms to be present in the Hamiltonian. Low-temperature series expansions, combined with a Pade approximant analysis are used to form estimates of the exponents. Strong evidence is found in support of potential sensitivity in the exponent beta , and examples of the specific heat exponent alpha ' are also discussed. The variability of the critical exponents is discussed in relation to the Universally postulate and to generalized lattice-gas models.

DOI
10.1088/0022-3719/7/22/006

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For pt. III see abstr. A23213 of 1974. The critical exponent beta is examined for two three-dimensional Ising model systems which allow admixtures of three and four-body terms to be present in the Hamiltonian. Low-temperature series expansions, combined with a Pade approximant analysis are used to form estimates of the exponents. Strong evidence is found in support of potential sensitivity in the exponent beta , and examples of the specific heat exponent alpha ' are also discussed. The variability of the critical exponents is discussed in relation to the Universally postulate and to generalized lattice-gas models.
For pt. II, see abstr. A64455 of 1973. The critical-point exponents of two Ising models with pure three-spin coupling terms are examined using low-temperature (T<T c ) series expansions for the thermodynamic functions. In particular the critical isotherm exponent delta is examined by applying numerical extrapolation methods to the critical isotherm. The two models are (a) a two-dimensional triangular lattice and (b) a three-dimensional face-centred cubic lattice. Direct numerical estimates of alpha ', beta and delta for (b) show that the scaling relations hold for this model and it is found that alpha '=0.655, beta =0.0538, gamma =1.24, nu =0.448, eta =-0.76, and delta =24 on the basis of the scaling relations. The numerical accuracy of the exponents beta and delta for model (a) does not permit an independent verification of the scaling relations; assuming scaling to hold here also, and using the exact result alpha '= 2 / 3 obtained by Baxter and Wu, it is found that gamma '=1.187+or-0.006, nu = 2 / 3 , eta =0.219+or-0.009, beta =0.073+or-0.003, and delta =17.3+or-0.8.
For pt. III, see abstr. A21890 of 1970. A new low-temperature expansion variable for the Ising model is given. Expansions in the new variable, x=1-tanh(J/kT), then converge right up to the critical point. Power series in the new variable are derived, and then analysed by the Pade method and the method of N-point fits. For the spontaneous magnetization of the Ising model in three dimensions. beta =0.312+or-0.002 for the simple cubic lattice, beta =0.3125+or-0.0015 for the body-centred cubic lattice, and beta =0.312+or-0.004 for the face-centred cubic lattice. For the low-temperature susceptibility gamma =1.30+or-0.02, gamma =1.28+or-0.04, and gamma '=1.29+or-0.04 for the simple cubic, body-centred cubic and face-centred cubic lattices respectively. For the low-temperature specific heat the authors estimate 1 / 16 <or= alpha =0.2 for all three lattices.
In this paper we reanalyze the series for the fluctuation of the order parameter, the fourth order fluctuation in the order parameter, and the specific heat for the spin [Formula: see text] model on three dimensional lattices. We find for the exponents, respectively, γ = 1.333 ± 0.001, γ2 = 4.725 ± 0.01, and α = 0.00 ± 0.04. We find no evidence for a correction to scaling term with exponent [Formula: see text] for the specific heat or for the susceptibility. Precise estimates for critical entropy and critical energy are obtained. From the critical amplitudes we obtain further support for the lattice–lattice scaling hypothesis and the one scale factor universality hypothesis.
High-temperature series are computed for a generalized three-dimensional Ising model with arbitrary potential. Three specific "improved" potentials (suppressing leading scaling corrections) are selected by Monte Carlo computation. Critical exponents are extracted from high-temperature series specialized to improved potentials, achieving high accuracy; our best estimates are gamma=1.2371(4), nu=0.630 02(23), alpha=0.1099(7), eta=0.0364(4), beta=0.326 48(18). By the same technique, the coefficients of the small-field expansion for the effective potential (Helmholtz free energy) are computed. These results are applied to the construction of parametric representations of the critical equation of state. A systematic approximation scheme, based on a global stationarity condition, is introduced (the lowest-order approximation reproduces the linear parametric model). This scheme is used for an accurate determination of universal ratios of amplitudes. A comparison with other theoretical and experimental determinations of universal quantities is presented.